Optimal. Leaf size=137 \[ \frac{7 a^{3/2} x^{5/2} \left (c-\frac{c}{a x}\right )^{5/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{5/2}}+\frac{a x^2 (18-a x) \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{5/2}}{3 (1-a x)^{5/2}}-\frac{2 x \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{5/2}}{3 \sqrt{1-a x}} \]
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Rubi [A] time = 0.154269, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6134, 6129, 98, 143, 54, 215} \[ \frac{7 a^{3/2} x^{5/2} \left (c-\frac{c}{a x}\right )^{5/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{5/2}}+\frac{a x^2 (18-a x) \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{5/2}}{3 (1-a x)^{5/2}}-\frac{2 x \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{5/2}}{3 \sqrt{1-a x}} \]
Antiderivative was successfully verified.
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Rule 6134
Rule 6129
Rule 98
Rule 143
Rule 54
Rule 215
Rubi steps
\begin{align*} \int e^{-\tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^{5/2} \, dx &=\frac{\left (\left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}\right ) \int \frac{e^{-\tanh ^{-1}(a x)} (1-a x)^{5/2}}{x^{5/2}} \, dx}{(1-a x)^{5/2}}\\ &=\frac{\left (\left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}\right ) \int \frac{(1-a x)^3}{x^{5/2} \sqrt{1+a x}} \, dx}{(1-a x)^{5/2}}\\ &=-\frac{2 \left (c-\frac{c}{a x}\right )^{5/2} x \sqrt{1+a x}}{3 \sqrt{1-a x}}-\frac{\left (2 \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}\right ) \int \frac{(1-a x) \left (\frac{9 a}{2}-\frac{a^2 x}{2}\right )}{x^{3/2} \sqrt{1+a x}} \, dx}{3 (1-a x)^{5/2}}\\ &=-\frac{2 \left (c-\frac{c}{a x}\right )^{5/2} x \sqrt{1+a x}}{3 \sqrt{1-a x}}+\frac{a \left (c-\frac{c}{a x}\right )^{5/2} x^2 (18-a x) \sqrt{1+a x}}{3 (1-a x)^{5/2}}+\frac{\left (7 a^2 \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}\right ) \int \frac{1}{\sqrt{x} \sqrt{1+a x}} \, dx}{2 (1-a x)^{5/2}}\\ &=-\frac{2 \left (c-\frac{c}{a x}\right )^{5/2} x \sqrt{1+a x}}{3 \sqrt{1-a x}}+\frac{a \left (c-\frac{c}{a x}\right )^{5/2} x^2 (18-a x) \sqrt{1+a x}}{3 (1-a x)^{5/2}}+\frac{\left (7 a^2 \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+a x^2}} \, dx,x,\sqrt{x}\right )}{(1-a x)^{5/2}}\\ &=-\frac{2 \left (c-\frac{c}{a x}\right )^{5/2} x \sqrt{1+a x}}{3 \sqrt{1-a x}}+\frac{a \left (c-\frac{c}{a x}\right )^{5/2} x^2 (18-a x) \sqrt{1+a x}}{3 (1-a x)^{5/2}}+\frac{7 a^{3/2} \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0613968, size = 87, normalized size = 0.64 \[ -\frac{c^2 \sqrt{c-\frac{c}{a x}} \left (\sqrt{a x+1} \left (3 a^2 x^2-22 a x+2\right )-21 a^{3/2} x^{3/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )\right )}{3 a^2 x \sqrt{1-a x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.159, size = 136, normalized size = 1. \begin{align*}{\frac{{c}^{2}}{6\, \left ( ax-1 \right ) x}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\sqrt{-{a}^{2}{x}^{2}+1} \left ( 6\,{a}^{5/2}{x}^{2}\sqrt{- \left ( ax+1 \right ) x}+21\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ){x}^{2}{a}^{2}-44\,{a}^{3/2}x\sqrt{- \left ( ax+1 \right ) x}+4\,\sqrt{a}\sqrt{- \left ( ax+1 \right ) x} \right ){a}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{- \left ( ax+1 \right ) x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a^{2} x^{2} + 1}{\left (c - \frac{c}{a x}\right )}^{\frac{5}{2}}}{a x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.46867, size = 680, normalized size = 4.96 \begin{align*} \left [\frac{21 \,{\left (a^{2} c^{2} x^{2} - a c^{2} x\right )} \sqrt{-c} \log \left (-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \,{\left (2 \, a^{2} x^{2} + a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \,{\left (3 \, a^{2} c^{2} x^{2} - 22 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{12 \,{\left (a^{3} x^{2} - a^{2} x\right )}}, -\frac{21 \,{\left (a^{2} c^{2} x^{2} - a c^{2} x\right )} \sqrt{c} \arctan \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \,{\left (3 \, a^{2} c^{2} x^{2} - 22 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{6 \,{\left (a^{3} x^{2} - a^{2} x\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a^{2} x^{2} + 1}{\left (c - \frac{c}{a x}\right )}^{\frac{5}{2}}}{a x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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